Tim Ting Chen, University of Southern California
Title: Inferring Domain-Domain Interactions from Protein-Protein Interactions
Understanding protein-protein interactions is one of the most important challenges in the post-genomic era. Behind protein-protein interactions there are protein domains physically interacting with one another to perform functions of proteins. Yeast 2-hybrid screens have generated more than 6000 interactions between proteins of the yeast Saccharomyces cerevisiae. This allows us to study the large-scale conserved patterns of interactions between protein domains. We use evolutionarily conserved domains defined in a protein-domain database called Pfam, and apply the Maximum Likelihood Estimation method to infer interacting domains that are consistent with the observed protein-protein interactions. Using these interacting domains, we predict about 100 novel yeast protein-protein interactions, among which many proteins have either unknown function or no observed interactions with other proteins. Our results can also be applied to predict protein-protein interactions of other species. (This is a joint research with Minghua Deng, Shipra Metah, and Fengzhu Sun.)
Dietmar Cieslik, University of Greifswald
Title: Steiner's Problem in Phylogenetic Spaces
A fundamental problem in many areas of classification is the reconstruction of the evolutionary past. Trees are widely used to represent evolutionary relationships. In biology, for example, the dominant view of the evolution of life is that all existing organisms are derived from some common ancestor and that a new species arises by a splitting of one population into two or more populations that not do not cross-breed, rather than by a mixing of two populations into one. Here, the high level history of life is ideally organized and display as tree. Trees may also be used to classify individuals of the same species. In historical linguistics, trees have been used to represent the evolution of language, while in the branch of philosophy known as stemmatology, trees may represent the way in which difference versions of a manuscript arose through successive copying. The principal of maximum Parsimony involves the identification of a combinatorial structure that requires the smallest number of evolutionary changes. It is often said that this principle abides by Ockham's razor1. Steiner's Problem is the "Problem of shortest connectivity", that means given a finite set of points in a metric space (X, ρ), search for a network interconnecting these points with minimal length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points, which are to be connected. Such points are called Steiner points. We will consider the problem of reconstruction of phylogenetic trees in the our sense of shortest connectivity. To do this we introduce so-called Phylogenetic spaces. These are metric spaces whose points are arbitrary words generated by characters (or letters; symbols) from some (finite) alphabet, and the metric measuring "sameness" of the words which is generated by a cost measure on the characters2. The "central dogma" is: A most parsimonious tree is a SMT in a desired chosen phylogenetic space. The following problems will be discussed:
In any case all these topics are problems in further research. ---------------------------------------------------------------- 1according to which the best hypothesis is the one requiring the smallest number of assumptions. 2or a similarity of the words generated by a scoring system.
- The duality of distance and similarity.
- Several combinatorial properties of phylogenetic spaces, and its finite languages, the sequence spaces. Particularly, Phylogenetic spaces are discrete metric spaces; that means: Each bounded subset is a finite one. Hence, an SMT exists for any finite set of given points (words) in the space, and it can be found by a suitable search-algorithm.
- Steiner's Problem is very hard in computational sense. It is at least ΝΡ-hard. But in specific cases it can be solved with simple and well-known methods.
- If we do not allow Steiner points, that means, we only connect certain pairs of the given points, we get a tree which is called a Minimum Spanning Tree (MST). The determination of an MST is simple. Consequently, we are interested in the greatest lower bound for the ratio between the lengths of these both trees: which is called the Steiner ratio. We look for estimates and exact values for the Steiner ratio in phylogenetic and sequence spaces. In any metric space the Steiner ratio is at least ½. Phylogenetic spaces achieves this ratio.
Miklós Csürös, Université de Montréal & Aleksandar Milosavljevic, Baylor College of Medicine
Title: Comparative physical mapping using Pooled Genomic Indexing 1. Introduction. Pooled Genomic Indexing (PGI) is a novel method for comparative physical mapping of clones onto known macromolecular sequences. PGI indexes genomes of new organisms using the already assembled genomic sequences of humans and other organisms. An application of the basic PGI method to BAC clone mapping is illustrated in Figure 1. PGI first pools arrayed BAC clones, then shotgun sequences the pools at an appropriate coverage, and uses this information to map individual BACs onto homologous sequences of a related organism. Specifically, shotgun reads from the pools provide a set of short (cca. 500 base pair long) random subsequences of the unknown clone sequences (100-200 thousand base pair long). The reads are then individually compared to reference sequences, using standard sequence alignment techniques  to find homologies. A set of homologies with close alignments in a reference sequence forms an index to the clone(s) identified by the set of pools the reads originate from. The indexes constitute a comparative physical map that enables targeted sequencing of regions of highest biomedical importance. Protocols for BAC pooling and shotgun sequencing of BAC pools have been originally developed at the Baylor Human Genome Sequencing Center in support of Clone-Array Pooled Shotgun Sequencing (CAPSS) . In contrast to PGI, CAPSS deconvolutes using fragment overlaps to produce an assembled sequence, and thus requires higher read coverage. 2. Pooling designs In addition to the simple arrayed pooling design shown in Figure 1, more sophisticated non-adaptive group testing methods  and clone-pool designs  can also be used with PGI. In general, a clone may be included in more than two pools, and the design does not necessarily correspond to an array layout. A homology between a reference sequence and a clone is detected by close local alignments between the reference sequence and shotgun reads originating from the pools the clone is included in. The success of indexing in the PGI method depends on the possibility of deconvoluting the local alignments. In the case of a simple array design, homology between a clone and a reference sequence is recognized by finding alignments with fragments from one row and one column pool. It may happen, however, that more than one clone are homologous to the same region in a reference sequence (and therefore to each other). This is the case if the clones overlap, contain similar genes, or contain similar repeat sequences. Appropriate pooling designs [5, 6] can ensure that not only individual clones, but clone sets of a certain size are also uniquely identifiable by the pools they are included in. Concentrating on a given index, classical probabilistic and combinatorial group testing results apply to PGI, the index being the analogue of a defective item in group testing terminology. Using standard techniques of modeling random shotgun reads , we analyze the probability of successful indexing for clones or clone sets, as well as probabilities for false positives and other aspects of index deconvolution. In contrast to a classical probabilistic testing problem, PGI generates many indexes at a time, and the probabilities of detecting homologies are determined by a controllable parameter in the experiment, namely, the number of random shotgun reads. Figure 1: The Pooled Genomic Indexing method maps arrayed clones of one species onto genomic sequence of another (5). Rows (1) and columns (2) are pooled and shotgun sequenced. If one row and one column fragment match the reference sequence (3 and 4 respectively) within a short distance (6), the two fragments are assigned (deconvoluted) to the clone at the intersection of the row and the column. The clone is simultaneously mapped onto the region between the matches, and the reference sequence is said to index the clone. 3. Experiments We tested the efficiency of the PGI method for indexing mouse and rat clones by human reference sequences in simulated experiments. The reference databases included the public human genome draft sequence, the Human Transcript Database, and the Unigene database of human transcripts. In one set of experiments we studied the efficiency of indexing mouse clones with human sequences. We selected 207 finished clone sequences. Pooling and shotgun sequencing were then simulated. We compared the performance of three pooling designs: a simple arrayed design, a random transversal design with two arrays, and a sparse array design (in which 2-sets of clones are uniquely identifiable). In contrast to the mouse experiments, PGI simulation on rat was performed using real shotgun reads from individual BACs being sequenced as part of the rat genome sequencing project at HGSC of Baylor College of Medicine. The only simulated aspect was BAC pooling, for which we pooled reads from individual BACs computationally. We selected a total of 625 rat BACs in the rat production database at HGSC. In both sets of experiments, using relatively few shotgun reads corresponding to 0.5-2.0 coverage of the clones, 60-90% of the clones can be indexed with human genomic or transcribed sequences. References:  Altschul, S. F., T. L. Madden, A. A. Sch¨aer, J. Zhang, Z. Zhang, W. Miller, and D. J. Lipman 1997. Gapped BLAST and PSI-BLAST: a new generation of protein database search programs. Nucleic Acids Res., 25:3389?3402.  Cai, W.-W., R. Chen, R. A. Gibbs, and A. Bradley 2001. A clone-array pooled strategy for sequencing large genomes. Genome Res., 11:1619?1623.  Du, D.-Z. and F. K. Hwang 2000. Combinatorial Group Testing and Its Applications (2nd ed.). Singapore: World Scientific.  Bruno, W. J., E. Knill, D. J. Balding, D. C. Bruce, N. A. Doggett, W. W. Sawhill, R. L. Stallings, C. C. Whittaker, and D. C. Torney (1995). Efficient pooling designs for library screening. Genomics 26, 21?30.  Kautz, W. H. and R. C. Singleton (1964). Nonrandom binary superimposed codes. IEEE Trans. Inform. Theory, 10:363?377.  D?yachkov, A. G., A. J. Macula, Jr., and V. V. Rykov 2000. New constructions of superimposed codes. IEEE Trans. Inform. Theory, 46:284?290.  Lander, E. S. and M. S. Waterman 1988. Genomic mapping by fingerprinting random clones: a mathematical analysis. Genomics, 2:231?239.
Ding-Zhu Du, University of Minnesota
Title: Some New Constructions for Pooling Designs
The non-adaptive and error-tolerance pooling design has gained an important application in DNA library screening. We propose two new classes of this type of pooling designs. These designs induce a construction of a binary code with minimum Hamming distance of at least 2d+2.
This work is joint with Hung Q. Ngo.
Dannie Durand, Carnegie Mellon
Title: Validating gene clusters
Large scale gene duplication, the duplication of whole genomes and subchromosomal regions, is a major force driving the evolution of genetic functional innovation. Whole genome duplications are widely believed to have played an important role in the evolution of the maize, yeast and vertebrate genomes. Two or more linked clusters of similar genes found in distinct regions on the same genome are often presented as evidence of large scale duplication. However, as the gene order and the gene complement of duplicated regions diverge progressively due to insertions, deletions and rearrangements, it becomes increasingly difficult to determine whether observed similarities in local genomic structure are indeed remnants of common ancestral gene order, or are merely coincidences. In this talk, I present combinatorial and graph theoretic approaches to validating gene clusters in comparative genomics.
Wen-Lian Hsu, Academia Sinica, Taipei, Taiwan, ROC & Wei-Fu Lu, National Chiao Tung University, Hsin-chu, Taiwan, ROC
Title: A Clustering Algorithm for Testing Interval Graphs on Noisy Data An important problem in DNA sequence analysis is to reassemble the clone fragments to determine the structure of the entire molecule. The idea is to describe each clone by some fingerprinting method. One such method is probe hybridization, in which a short molecule called a probe is used to bind (or hybridize) to the clone, the hybridization data is then used to determine the overlap relationships of the clones [1,5,10]. An error-free version of this problem can be modeled as an interval graph recognition problem, where an interval graph is the intersection graph of a collection of intervals (we shall use the term interval and clone interchangeably in this abstract). Several linear time algorithms have been designed to recognize interval graphs [2,7,8]. However, since the data collected from laboratories almost surely contain some errors, traditional recognition algorithms can hardly be applied directly. It should be noted that there is no obvious way to modify existing linear time interval graph test algorithms to incorporate small errors in the data. That is, a single error could cause the construction of the interval model to halt. Several related approximation problems have also been shown to be NP-hard [3,4,9,12]. To cope with this dilemma we propose a clustering algorithm that is robust enough to discover certain types of input errors and to correct them automatically. The errors considered here include false positives (two originally non-overlapping intervals appear to overlap in the data) and false negatives (two originally overlapping intervals appear not to overlap in the data). The kind of error tolerant behavior considered here are similar in nature to algorithms for voice recognition or character recognition problems. Thus, it would be difficult to guarantee that a given error tolerant algorithm always produces a desirable solution; the result should be justified through benchmark data and real life experiences. Our clustering algorithm has the following features: 1. the algorithm will assemble the clones correctly when the data is error-free. 2. in case the error rate is small (say, less than 10%) the test can likely detect and automatically correct the false positives and false negatives. 3. the test can accommodate some probabilistic assumptions about the overlapping relationships. 4. by providing confidence score on the endpoints, the test will also identify those parts of the data that are problematic, thus allowing biologists to perform further experiments to clean up the data. The idea of our error-tolerant algorithm is based on an algorithm modified from a linear time test of interval graphs . In the algorithm, we get rid of the rigid structure such as PQ-trees  or sophisticated ordering [7,8] that are powerless in dealing with errors. Instead, we make use of a certain robust local structural property to weed out false positives that are remotely related to the current interval first. We then use this structural property to treat local false positives, false negatives and to order the neighboring clones. Such a structural property has to do with the overlapping relationships among neighbors of an interval, which are described below. Assume we have an interval model for a given graph. For each interval u, consider those intervals that strictly overlap u (do not contain u nor are contained in u). Divide these intervals into two parts: A(u), those passing through the left endpoint of u and B(u), those passing through the right endpoint of u. The adjacency relationships between intervals in these two sets satisfy a monotone triangular (MT) property. In case the data contains a reasonable small percentage of error (say 10%), it would still be possible to recover such a MT property approximately. This phenomenon has been illustrated in our result for testing the consecutive ones property of a (0,1)-matrix under noise . In order to accommodate false positives and false negatives in the data, some clustering and thresholding techniques are adopted. Although it is hard to guarantee that the algorithm produces an approximate solution model within a certain percentage of error, the experimental result on synthetic data is extremely convincing that the algorithm should give satisfactory interval models in most cases. Our techniques are quite flexible in that we can even accommodate the case where the overlapping relationships are expressed in terms of probabilistic values instead of the deterministic zeros and ones. At the end, the algorithm could provide the confidence score for each endpoint position. This could prompt biologists to redo some experiment to clean up the noisy part. Another important feature of our algorithm is that, although it utilizes the local monotone triangular property, the error-correcting effect can be quite global. When it appears that a clone creates a disrupting behavior for its neighbors, this clone could be deleted from further consideration. Such a measure is installed to prevent possible disastrous result. In summary, we have modified previous rigid algorithms for testing interval graphs into one that can accommodate clustering techniques as well as probabilistic assumptions, and produces satisfactory approximate interval models for most synthetic data. References: 1. F. Alizadeh, R.M. Karp, L.E. Newberg, and D.K. Weisser, (1995) Physical mapping of chromosomes: a combinatorial problem in molecular biology. Algorithmica, 13, 52-76. 2. K.S. Booth and G.S. Lueker, Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-tree Algorithms, J. Comput Syst. Sci. 13, 1976, 335-379. 3. M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980. 4. M.C. Golumbic, H. Kaplan and R. Shamir, On the Complexity of DNA Physical Mapping, Advances in Applied Mathematics 15, 1994, 251-261 5. M.T. Hallett, (1996) An Integrated Complexity Analysis of Problems from Computational Biology. Ph.D. thesis. Department of Computer Science, University of Victoria, Victoria, BC, Canada. 6. W.L. Hsu, A simple test for the consecutive ones property, Journal of Algorithms 42, 1-16, 2002. 7. W.L. Hsu, A simple test for interval graphs, LNCS 657, 1992, 11-16. 8. W.L. Hsu and T. H. Ma, Substitution Decomposition on Chordal Graphs and Applications, LNCS 557, 1991, 52-60. 9. H Kaplan, R Shamir, and R.E Tarjan, (1994) Tractability of parameterized completion problems on chordal and interval graphs: minimum fill-in and physical mapping. In FOCS'94. 10. J.D Kececioglu, (1991) Exact and Approximation Algorithms for DNA Sequence Reconstruction. Ph.D. thesis. Technical report TR 91-26, Department of Computer Science, University of Arizona. 11. W.F. Lu and W.L. Hsu, A Test for the Consecutive Ones Property on Noisy Data, submitted to Journal of Computational Biology. 12. M. Yannakakis, Computing the Minimum Fill-In is NP-Complete, SIAM J. Alg. Disc. Meth 2, 1981.
Frank Hwang, National Chiao Tong University
Title: Random pooling designs with various structures
Typically, there are two ways to construct pooling designs, one through using some mathematical structure, and the other through randomness. Recently, several constructions which combine both the structure and randomness elements have been proposed. These include the random k-set design, the distinct random k-set design, the random k-size design and several designs by Macula. We will first update the best formulas to compute the success probabilities of identifying a positive clone with the first three models.
For the basic 1-stage Macula's design with parameters (n,k,d), Macula gave an approximate algorithm to identify positive clones from this design, and an analysis of the success probability. For d=2, which is the focus of Macula's analysis, we will propose a new method which provably identifies more positive clones, but the analysis is much more difficult. So far, we have exact result only when the number of positive clones is at most 3.
Macula also gave a 2-stage design and an approximate formula with 2 summation signs, and a further approximation in explicit form. We will gave an exact formula also with 2 summation signs, and an approximation in explicit form, which is more accurate than Macula's explicit approximation. The purpose of the approximation is to facilitate the finding of optimal design parameters.
Finally, we will provide some numerical comparisons between the various random models, and between Macula's analysis and our new analysis.
F.K. Hwang and Y.C. Liu Title: Error-tolerant Pool Designs with Inhibitors Pooling designs are used in clone library screening to efficiently identify positive clones from negative clones. Mathematically, a pooling design is just a nonadaptive group testing scheme which has been extensively studied in the literature. In some applications, there is a third category of clones called ''inhibitors'' whose effect is to neutralize positive clones. Namely, the presence of an inhibitor in a pool dictates a negative outcome even though positive clones are present. Sequential group testing schemes, which can be modified to 3-stage schemes, have been proposed for the inhibitor model, but it is unknown whether a pooling design (a one-stage scheme) exists. Another open question raised in the literature is whether the inhibitor model can treat unreliable pool outcomes. In this paper we give a pool-design (one-stage), as well as a 2-stage scheme for the inhibitor model with unreliable outcomes. The number of pools required by our schemes are very comparable to the 3-stage scheme.
Esther Lamken, Caltech Title: Two applications of designs in biology In this talk, I will describe two applications of combinatorial designs in biology. The first application uses ideas from finite geometry and design theory to help design pooling experiments. The second application extends work of Sengupta and Tompa. I will describe how ideas from design theory and cryptography can be used to deal with the problem of multiple step failure in the manufacture of DNA microarrays at Affymetrix.
Wen-Hsiung Li, Department of Ecology & Evolution, University of Chicago
Title: Methods for Aligning Long Genomic Sequences
Many eukaryotic genomes will soon be completely or near completely sequenced. An extremely challenging problem is how to align genomic sequences, which is usually the first step in comparative analysis. Alignment of genomic sequences is much more difficult than alignment of protein sequences or DNA sequences of protein coding regions because (1) the large demand of computer time and memory space, (2) the sequences often have been scrambled by insertion of transosable elements, translocation, and inversion, and (3) the existence of highly divergent regions. Some methods for aligning genomic sequences will be presented.
Alan C.H. Ling, Charles J. Colbourn, and Martin Tompa; University of Vermont
Title: Construction of Optimal Quality Control for Oligo Arrays
Oligo arrays are important experimental tools for the high throughput measurement of gene expression levels. During production of oligo arrays, it is important to identify any faulty manufacturing step. We describe a practical algorithm for the construction of optimal quality control designs that identify any faulty manufacturing step. The algorithm uses hillclimbing, a search technique from combinatorial optimization. We also present the results of using this algorithm on all practical quality control design sizes.
Peng-An Chen and F. K. Hwang, Department of Applied mathematics, National Chiao-Tung University, Hsinchu, Taiwan 30050 R.O.C.
Title: On a Conjecture of Erdos-Frankl-Furedi on Nonadaptive Group Testing
d-disjunct matrices become the dominating tool in studying nonadaptive group testing, which has attracted a lot of attention due to its recent application to pooling designs in DNA mapping. By testing the items one by one, clearly we can identify n items in n tests. Huang and Hwang(1999) raised the fundamental question what is the maximum n such that a d-disjunct matrix has at least n rows? Namely, let N(d) denote this maximum n for given d. Then for all n £ N(d) the individual testing algorithm is the best. Erdos, Frankl and Furedi(1985) studied the same problem in their study of extremal set theory. They conjuctured that N(d) ³ (d+1)2 and claimed they can prove for d = 1,2,3. Huang and Hwang(1999) provided the proof for d = 1,2,3 and also claimed that their method also works for d = 4, but it would be too messy to produce the details. We propose a new approach to attack the Erdos, Frankl and Furedi conjecture. Instead of working diligently on many subclass, and then piece them together to produce the proof for one d-value, we study the properties of the d-disjunct matrix at some critical point, namely, when t = d(d+2) and the d-disjunct matrix is minimal under some reduction operations. These properties provide a powerful d = 4 case follows immediately, and d = 5 case can be proved elegantly.
Anthony J. Macula, SUNY Geneseo
Let [t] represent a finite population with t elements. Suppose we have an unknown d-family of k-subsets C of [t]. We refer to C as the set of positive k-complexes. In the group testing for complexes problem, C must be identified by performing 0, 1 tests on subsets or pools of [t]. A pool is said to be positive if it completely contains a complex; otherwise the pool is said to be negative. In classical group testing, each member of C is a singleton. In this talk we discuss adaptive and nonadaptive algorithms that identify the positive complexes. We also discuss the problem in the presence of testing errors. Group testing for complexes has been applied to DNA physical mapping and has potential application to data mining.
Fred Roberts, Rutgers University - Piscataway, NJ, USA
Title: Consensus List Colorings of Graphs and Physical Mapping of DNA
Abstract: In graph coloring, one assigns a color to each vertex of a graph so that neighboring vertices get different colors. We shall talk about a consensus problem relating to graph coloring and discuss the applicability of the ideas to the DNA physical mapping problem. In many applications of graph coloring, one gathers data about the acceptable colors at each vertex. A list coloring is a graph coloring so that the color assigned to each vertex belongs to the list of acceptable colors associated with that vertex. We consider the situation where a list coloring cannot be found. If the data contained in the lists associated with each vertex are made available to individuals associated with the vertices, it is possible that the individuals can modify their lists through trades or exchanges until the group of individuals reaches a set of lists for which a list coloring exists. We describe several models under which such a consensus set of lists might be attained. In the physical mapping application, the lists consist of the sets of possible copies of a target DNA molecule from which a given clone was obtained.
Chuan Yi Tang, National Tsing Hua University Title: Computational Gene Identification by Comparative Genomics Approach The problem of gene identification is one of the most fascinating problems in recent bioinformatics research area. Due to the large amount DNA sequencing data every day in laboratory; scientists are looking for faster and more accurate computational methods to recognize gene structure. Most of the existing gene prediction methods only consider a single genome. The strategies are combined with statistics, special signal sequences (such as splice sites or promoter), and with homology information in databases. Here we propose different methods, which apply comparative analysis between related species. We can identify the important functional regions, which will tend to be conserved during evolution process between some human and mouse genomic sequences.
Lusheng Wang, City University of Hong Kong Title: Approximation Algorithms for Distinguishing Substring Selection Problem Consider two sets of strings, B (bad genes) and G (good genes), as well as two integers db and dg (db £ dg). A frequently occurring problem in computational biology (and other fields) is to find a (distinguishing) substring s of length L that distinguishes the bad strings from good strings, i.e., for each string si Î B there exists a length-L substring ti of si with d(s, ui) £ db (close to bad strings) and for every substring ui of length L of every string gi Î G, d(s, ui) ³ dg (far from good strings). We present a polynomial time approximation scheme to settle the problem, i.e., for any constant e > 0, the algorithm finds a string s of length L such that for every siÎ B, there is a length-L substring ti of si with d(t, is) £ (1 + e)db and for every substring ui of length L of every gi Î G, d(u, is) ³ (1 - e)dg, if a solution to the original pair (db £ dg) exists. Since there are a polynomial number of such pairs (db, dg), we can exhaust all the possibililities in polynomial time to find an good approximation required by the correpsonding application problems.
Chih-wen Weng and Tayuan Huang, National Chiao Tung University Title: Constructions of Non-adaptive Pooling Designs We show how to construct non-adaptive pooling designs from a ranked atomic (meet) semi-lattice. All these pooling designs are e-error detecting for some e which can be chosen to be very large compared to d, the maximal number of defective items(or positives). Eight new classes of non-adaptive pooling designs are given, which are related to the Hamming matroid, the attenuated space, and six classical polar spaces. The general construction of ranked atomic semi-lattices from known ones are discussed.
Jer-Shyan Wu, Department of Computer Science & Information Engineering, Chung Hua University, Hsinchu, TAIWAN Title: Efficient Algorithms for Weighted Genome Rearrangements Given two different genomes with the same numerical labels, the goal of genome rearrangements is to find the minimum number of reversal operations that leads from one genome to the other. Consider the biological reality, each reversal has different evolutionary time and probability, i.e., each weight is distinct not identical. In this paper, We discuss the simple weighted model : each reversal weight is portional to its length, and try to propose some efficient algorithms to solve this probelm.